Is #f(x)=(2x^3-7x^2+13x-11)/(x-3)# increasing or decreasing at #x=2#?

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Answer 1 · 2018-07-15T12:31:49

#f(x)# is monotonously decreasing in the point #x=2#

#(u/v)'=(u'v-uv')/v^2#
By this rule we get

#f'(x)=((6x^2-14x+13)(x-2)-(2x^3-7x^2+13x-11))/(x-3)^2#

expanding the numerator

#f'(x)=(6x^3-14x^2+13x-18x^2+42x-39-2x^3+7x^2-13x+11)/(x-3)^2#

collecting like Terms

#f'(x)=(4x^3-25x^2+42x-28)/(x-3)^2#

#f'(2)=(32-100+84-28)/(-1)^2=-12<0#